Potential flow about two-dimensional hydrofoils

ARCHEF

J. Fluid Mech. (1987), voi, 28, part 1, pp. 113-129 Pri,Ued in Great Britoin

Potential flow about two-dimensional hydrofolls

By JOSEPH P. GIESING AND A. M.

_{0. SMITH}

Douglas Aircraft Company, Aircraft Division, Long Beach, CaJifornia (Received 15 November 1965 and in revised form 6 June 1966)

This paper describes a very general method for determining the steady

two-dimensional potential flow about one or more bodies of arbitrary shape operating

at arbitrary Froude number near a free surface. The boundary_{condition of zero}

velocity (solid wall) or prescribed velocity (suction or blowing) normal to the body

surface is satisfied exactly, and the boundary condition of constant pressure on

the free surface is satisfied using the classic small-wave approximation. Calcula-tions made by the present method re compared with analytic results, other

theoretical calculations and experimental data. Examples for which no com-parison exists are also presented to illustrate the capability of the method.

1. Introduction

The pressure distribution and flow field about hydrofoils or systems of

hydro-foils is important for several reasons: the determination ofcavitation inception, the calculation of boundary-layer characteristics, and the determination of the

inviscid hydrodynamic characteristicslift,_{moment, and wave resistance. Thin}

hydrofoil theory (Keldyach & Lawrentjew 1935; Walderhaug 1964) predicts pressures that are physically unrealistic at the hydrofoil _{nose (usually infinite)} and unusable either for the determination of cavitation or for the calculation of

boundary-layer characteristics. Other theories, some based on

conformal-mapping techniques (Havelock 1936; Kochin 1937; Nishiyama _{1957) are}

re-stricted to single bodies. Two theories that do consider the flow about more than one body are given by Coombs (1950) and Isay (1960), but both of these_{are only}

approximate.

This paper presents a method that determines the flow_{about one or more}

large-aspect-ratio hydrofoils, of arbitrary shape, moving with constant forward speed and at arbitrary Froude number. The Neumann boundary condition on the surfaces of the hydrofoils, that is, zero velocity (solid wall) or prescribed velocity normal to solid boundaries (suction_{or blowing), is satisfied exactly by} a surface source distribution that also satisfies the classic small-perturbation

free-surface boundary condition.

The techniques on which the present method is based (Smith _{& Pierce 1958)}

are not restricted to two-dimensional flow, and therefore an extension to include three-dimensional problems such as the flow about submarines and ships may be possible (see also Hess & Smith 1964). In addition, it seems possible to refine the

method to include higher order terms in the free-surface boundary _condition.

8

Fluid Meoh. 28

1gb: y.

Tech& sche

Hcbi

114 Jo8eph P. Gie8ing and A. M. 0. Smith

2. DescrIption of the problem

The problem of concern is the upoavitated two-dimensional potential flow past one or more bodies moving with constant forward speedbeneath a free surface. Since the potential-flow model is ued and surface tension effects neglected, the governing equations of fluid motion and pressure are the Laplace and Bernoulli

equations, respectively. They are

V2b=O,

p/p+V.V+gy=p4p+U,

with V=V(b.

The symbols V, ,p,p and g are the velocity, potential, pressure, mass density and

acceleration due to gravity, respectively and the subscript refers to infinity upstream at the free surface.

Free Surface.

U,,)

y

Pzava 1. Schematic diagram.

The fluid is assumed to occupy the lower half-plane and to be bounded by a free surface upon which the pressure is constant. Figure 1 shows the co-ordinate system and direction of fluid motion. Boundary conditions must be applied to the surfaces S of the bydrofoils and to the free surface. The boundary condition applied at the hydrofoil surfaces requires that the velocity normal to the surfaces either be zero (solid body) or be prescribed (suction or blowing). After the

intro-duction of the disturbance potential q, defined by

(3) the boundary condition on the hydrofoil surfaces can be written

= - U,,i.n +f(S) on 5, (4)

where S defines the hydrofoil surfaces, n the outward normal vector andf(S) the

blowing or suction velocity.

The boundary condition on the free surface requires that the pressure be

constant and that the free surface y = (x) be a streamline. It can be shown

that

the resulting boundary condition is exactly

Ø2 ₁

[u4]

_[

(v+)

+-j

x

8-2

Pot ential flow about two-dimen8ional hydrofoils 115

Thu condition is non-linear in the potential ç5, and it is applied to a surface that is not known a priori. The last boundary condition that must be applied expresses

the physically observed fact that the disturbance velocities due to the ensemble of hydrofoils vanish only far upstream and far below the foils, that is,

Vç-*0

when x-,oo,

Vq5-*0 when y-.

(6)

A disturbance downstream (x> 0) is observed in the form of trailing surface

waves that apparently would extend to infinity were it not for the effects of viscosity.

Simplification of the free.surface boundary condition

The problem as posed is highly intractable, because of the free-surface boundary

condition (5). Since this boundary condition is non-linear and is applied to a surface that is not known, some simplification of this boundary condition is needed. A linear boundary condition applied at the undisturbed surface y =0 is

obtained from (5) if the wave heights and disturbance velocity at the free surface

are assumed to be small compared to the characteristic lengthe (usually body

chord length) and the forward speed U, respectively. (5) then becomes

ø2ç5/x2+v(q/by) =0 on y= 0, (7)

where v = g/ U. With these same assumptions, (2) furnishes an expression for the linearized wave height. Thus

(8)

(7) is also obtained as the lowest order term in an expansion of (5) when q and

are expanded in terms of some small parameter (Wehausen & Laitone 1960). The only requirement for e is that çb vanish in the region of the free surface when

e vanishes. It is proposed here to consider this expansion in terms of /h where

his hydrofoil submergence depth. The first term of the expansion is to satisfy the

boundary condition on the body surface exactly, that is, satisfy (4). The rest of

the terms are to satisfy a homogeneous boundary condition on the body surface,

that is, satisfy (4) with the right-hand side set equal to zero.

If the depth is sufficiently large, terms of higher order than e can be retained,

which increases the accuracy. For the present, only the first-order term (7) is used, since it offers a practical solution for all but extreme oases and presents no convergence difficulties.

In what is to follow the Oharacteristic non-dimensional parameter is the Froude number. The Froude number is defined in (9) as follows

Fr =

U(gt)4.

(9)

3. Solution

The problem to be solved is reduced to finding the solution of the Laplace

equation that is consistent with the boundary conditions given by (4), (6) and (7)

and which satisfies the Kutta condition on each body, if applicable. First, an

elementary solution of the Laplace equation is assumed, of the form G(P,Q)= ln[r(P,Q)]+K(P,Q).

e....- -.

116 Jo8eph P. Giesing and A. M. 0. Smith

The first term on the right is the familiar potential at P due to a unit source at Q. The second term is a function, non-singular in the region of interest, that causes

G(P, Q) to satisfy the free surface, radiation andfar-field boundary conditions.

The radiation and far-field boundary conditions are given by (6). The function G(P, Q) will be called a source because of its close relation to the usual simple

source. Starting with Green's third identity and using the source G(P, in place

of the usual source potential, the following equation can be obtained (see Lamb 1932, p. 60),

1

o(q)G(P,q)dS; (10)

2ir s

that is, .b(P) is the potential at a general point P due to a distribution, on the surface 2, of sources of strength o-(q). The value of Q on the body surfaces is

called q.

Since an expression for the potential, (10), is known, the boundary condition on the body surface, (4), may be written as follows

= Uc,i.n(p) = urn

o(q)VG(P,q).n(P)dS, (11)

-- øn p_,p2iTs

with the assumption of no flow through the hydrofoil surface (suction Or blowing). Here p is a point on S. If the limit is taken in (11) and the principal part abstracted

from the integral, then the result is a non-singular Fredholm integral equation

of the second kind for the unknown source strength o. For convenience the limit will not be taken and the integral of (11) will be left in its present form.

(11) then insures that the Neumann boundary condition (4) is satisfied. The function K(P, Q) must now be found such that G(P,Q) satisfies the free-surface

condition (7) and the radiation condition (6). Note that if the individual source potential G(P,Q) satisfies (6) and (7), an entire distribution of such sources will

also satisfy (6) and (7), since these boundary conditions are linear and homo-geneous. From Kochin, Kibel & Roze (1964), we obtain K(P, Q) as

.K(P,Q) = Re [in

(z)

2e_iv3f

__dt]

(12)

where z and c are the complex co-ordinates of P andQ,respectively, and the bar

indicates the conjugate. It is convenient at this point to introduce the complex

potential F such that G = Re (F) and dF/dz = aG/ax - i G/y. Here F is

F(z,c)

ln(z_c)+1n(z_)_2e"f

dt.

(13)

The integral in (13) can be put into more convenient form by transformation

using the relation

_{v(tz) = k(zc)}

-and then separating the principal part. The result is r

F(z,c) = ln(zc)+ln(z)+2PV

Jo A;V

I (14)

where PV means the principal value. Introduction of the real part of (14) into (11) leads to an integral equation for the unknown source distribution o- that satisfies

Pot ential flow about two-dimensional hydrofoil8 117 the Neumann boundary condition on the body surface (4), the linearized free-surface condition (7), and the radiation condition (6). Only the Kutta condition

is left to consider.

To satisfy the Kutta condition on each of the bodies, additional solutions must

be obtained that satisfy the condItions (6) and (7) but are homogeneous in the

Neumann boundary condition (4), that is, solutions where ôçb/øn = 0 on S. Any

number of these solutions may be added to the solution given by (11) without

violating (4). Specifically, one such solution is needed for each body to which a Kutta condition is applied. The only non-trivial solution that satisfies these con-ditions and is also non-singular within the fluid is a circulatory-flow solution about

any one of the bodies. The circulation or cyclic constant is arbitrary. One such solution must be obtained for each body that can maintain a circulation. The

cyclic constants associated with the bodies are adjusted, since they are arbitrary,

until the Kutta condition on each body is satisfied.

A solution that produces a circulation about a particular body is obtained by

placing a vortex or distribution of vorticity within the body to produce the desired circulation and by cancelling the resulting flow normal to the body surface with a source distribution. The vortex or distributionof vorticity must, of course, satisfy the boundary conditions (6) and (7), just as the source distribution

satisfies it. The potential for such a vortex is

jr I

1

= Re_Iln(z_c)_ln(z)-2PV I

dk+2iie"(I,

(15)

21TL

Jo iW

j

where r is the circulation or cyclic constant. If c6r/0 denotes the velocity

generated by the gradient of (15) normal to and at the body surface, then the source distribution °r that cancels this velocity is given by the integral equation

(11) with Uc0.fl(p) replaced by aqr(p)/an.

Solution of the integral equation

The integral equation to be solved, (11), can be written

= urn -'-f o(q)VG(P,q).n(P)dS,

(16)

s

where can represent either i. n orøçb/c9n, and G(P, Q) is given by the real

part of (14). For blowing or suction cases, f(S) must be added to U. i. n. If the

integral in (16) is split into a sum of integrals whose range of integration, AS, is

small enough so that source strength, o, can be assumed constant across that range, then (16) can be written approximately as

N rSk+SkJ2

-

urn -

o VO(P,q).n(P)dS. (17)

p+p27Tk=1 JSk-Sk/2

There are now N unknown source strengths 0k To obtain N equations, (17) is made to hold at N points on the body surface. The points selected are the

mid-points of the elements, that is, p5. The surface co-ordinates of p5 are S. (17) then becomes

N 1 I CSt+SkI2 1

-

=

118

Joaep E.

tnd A. M. 0. S-mith

The first halF of (18) reprpnt8 !f equations in N unknowns and may be solved

using stand&rd matrix t&hniques once the influence coefficients Aik are known.

To be consistent with the approximation of constant source strength over the

element, the element is approximated by a flat segment (see figure 2). The above approximations become exact in the limit as N - . Toenable thelimit indicated

in (18 be taken, the integration is performed analytically for arbitrary P. In

I.

Free surface

z.

Uc0

kth element Body surface

Fiomx 2. Typical fiat-surface source element and its reflexion.

performing this integration it is convenient to use Oomplex variables; specifically, the complex potential F is used in place of 0. The following relations hold between

the vector and complex forms:

V0 = 1(Re dF/dz) +j( ImdF/dz),

= i(Re 4) +j(Im 4),

(19)

where I represents the integral found in (18) and 'k is its complex equivalent.

If only the kth element is considered, the subscript k may be dropped and I may

be written as _{S+S/2 dE}

i=1

dS.

(20) dz

From figure 2 the following relations are evident

dS = dce = de(A. (21)

By means of (21) and (14), (20) becomes

]

- 2-i[ex) -

ez))}.

(22)

= eln

La - c \ I

Ia - \

In (-= I

2 e?

₊

I_{I PV I}

r

L Jo

dk PV I

r e&

-\ZC2J

kv

-

_Jo

k-v

c1 de 1' 1 ₁

I' ike')

'I

I = e

I

- +

e"

I + 2PV I

dk - 2irv e--) d

Jc1 zc

jo izc

Jo

kv

Reflected body

-

_-

= r + im

Potential flow about two-dimensional hydrofoil8 119

The integrals left to evaluate, which are of the form

I'

PVI

die,

Jo kv

canbe handled by using contour integration in the qomplex k-plane. With refer-ence to figure 3, the desired integral can be written

e0

'

' ez

PYI

dk=

I +1

dk=I

-I

die (23)

Jo kv

j'

j

kv

J2 J4

j

kv

im

1 3

FIGuRE 3. Integration contour in the complex k-plane.

since the total contour encloses no singularities. It is possible to find an angle_ft such that the exponent of is purely real along contour 5 and such that the

integral along contour 4 vanishes. Thus it is found that

ft=tan_1(---\y+y

with. = iy and z

x+iy. Note that the sign offtis determined by the sign of - (x - ), since both y and y are negative.

The integral along ccntour 2 is known once the sign of ftis knowti,'

r

I

dk = (sgnft)iie.

(24)

J2 kv

S

The integral along contour 5 is evaluated when the exponential is real, that is, with k = r(1 +itanft), which gives

e

kv

dk =f

i()du = e"()E1[iv(z--)],

(25)

where E1 is the exponential integral. Hess & Smith (1966) present a rapid and accurate method for the evaluation of the exponential integral. Here

J(y+y)2+(x)2

1.

(!'+v)

Substituting the expressions for the integrals in (24) and (25) into (23) gives PV

die = (sgnft)iie_+èE1[iv(zc)].

(26)

120 Joseph P. Oie8ing and A. M. 0. Smith

Substituting the value of the integral in (26) into (22) completes the evaluation

of I. (See Smith, Giesing & Hess 1963 for further details.)

I = e

In + eU

{In (.L)

+2 ei) E1[ iv(z

-The quantities a, c1 and c2 are shown in figure 2. -The expressiOn for I in (27) may

be used in (18) with the aid of (19) to produce an expression for the influence coefficients Aik.

Velociti, and pressure field

The velocity or the potential field is easily found once the source distribution over

the bodies is known. The disturbance velocity at any point P due t'b a source distribution is obtained by taking the gradient of (10). The onset flow must be added to this disturbance field to produce the total velocity field. Let the

sub-script 0 denote circulation-free flow and the subsub-script 1' denote pure circulatory flow. Then for circulation-free flow the total velocity is

V0(P) = V0 = V(Ux+ç50) =

ui+fo.o(q)vo(P)ds.

(28)

For a pure circulatory flow the velocity field is

Vr(P) =

r{vcsr+fo.r(q)vG(P)ds}.

(29) where T is the circulation or cyclic constant. The integrals of (28) and (29) are evaluated with the same approximations as the integrals of (16). Therefore

r N

I o(q)VG(P,q)dS

oI(I1,

(30)

JS

where I is given by (19) and (27). As was mentioned before, there is one circu-latory flow associated with each body which causes a circulation about that body

alone. A combination of these circulatory flows with the non-circulatory flow produces the total velocity field

V(P) = V0(P) + FiVri(P) + rsvr(P) +... + rRVrR(P), (31)

where B is the number of bodies that can maintain a circulation. The cyclic

constants r1, F2, ..., r are adjusted to satisfy the Kutta condition at each of the

hydrofoil trailing edges by requiring that the velocity at the trailing edge elements, i.e. the one above and the one below, be equal. The point P is arbitrary

and need not be on the body surfaces. Thus the entire flow on or off the body

surface is determined. where

2

'

= 1i,

E1[ - iv(z - c2)] - 4iii [8 ei) 82

e_i.)

(27)

f0,fl>0

(z upstream of c1),

1,fl1<0

(z downstream of c1), -fO,fl2>0 (z upstream of c2),

- Present method

60 o Experimental data

50

3

8°

Fiovrtx 4. Comparison of calculated and experimental pressure distributions on a NACA 23012 airfoil with fixed slot and slotted flap at 8° angle of attack. CL = 226.

As an example of a hydrofoifat infinite depth, figure 4 shows a comparison of

the calculated and experimental pressure distribution on an airfoil with a fixed slot and slotted flap. The experimental data were obtained by Harris & Lowry

(1942). The agreement is good except over a part of the lower surface of the slot where the flow is apparently separated. In all cases the force coefficients shown

are those calculated by the present method.

flow gboug gwo-dimenaonvJ hyc$rofoila 121

The pressire can be obtained £rm . 4. pressure coefficient is defined as

follows: ₌

_{= 1 V.V/U,}

₍₃₂₎

wher he term pgy is purely hydrostatic. The forces and moments are easily found after the pressure is known by direct integration as follows:

CR=fCPn.idS

(33a)

CL=f_CPfl.idS

(33b)

Cm=fC(nxr)dS

where r is the position vector from the moment reference point to a surface point.

These integrals are evaluated approximately using the trapezoidal rule, that is,

summing element by element.

-4. Calculated results

_{Special cases}

Two special cases of the theory exist when the hydrofoil operates at an infinite

depth or when it operates at a Froude number of zero. These cases correspond to

an airfoil in an nurestricted fluid and in ground effect, respectively. They are

special because the non-linear boundary condition on the free surface is satisfied

Jo3ep 4. iesng and A. M. 0. Snith

As an example Of the su pecial case, figure 5 presents a comparison of the

analytic and calculated pressure distribution on a hydrofoil operating at a Froudé

number of zero. The analytic solution was developed by Giesing. (1968). The

agreement is good.

U

WAUUU

I. Analytic Present method Lower surface 0154 I FIGURE 5 30 20 10 0 20 10 0

FIGURE 5. Comparison of analytic and calculated pressure distributions on a hydrofoil

opera-tingatFr=0. CL=-211.

FIGtrltE 6. Comparison of lift and drag coefficients of a circular cylinder as calculated by the present method and as calculated by Havelock. Results for a dipole are also shown.

Compari8on with other method8 and experimental data

One of the first calculations of the forces on a body under a free surface was for a circular cylinder. Ilavelock (1928) derived expressions for the lift and drag of

a dipole under a free surface, which were assumed to apply to a circular cylinder. Later (1936), he presented the exact solution for the circular cylinder. Figure 6 is

a comparison of the wave-resistance coefficient, CR, andlift coefficient, CL, of a circular cylinder, as calculatod by the present method and as given by Havelock,

or a dipole as given by ilavolock. The lift and dragcoefficients are defined

as the lift and drag divided by p U t/2 where t, in this case, is the circle radius.

The number of defining elements used for the calculation by the present method

is 30 and the submergence depth is two radii.

The effect on the lift and drag of the number of elements used to define the circle is of interest, since the accuracy of the present method is a function of this

num-ber. The following table presents values of lift and drag coefficients at a Froude

number of FO for circles described by 30, 60, 120 and 240 points and exact values

calculated by Havelock (1936). The table shows a maximum error of 1.4% for wave resistance but a slow convergence to higher accuracy as the number of points increases. The values of lift coefficient calculated by the present method,

o Present method - Havelock (exact) -- Havelock (dipole)

jl

4

10

0 2 3 U (g74 FIGuRE 6

Number of

defining Havelock?

elements 30 60 120 240 exact

2-1082 20671 2-0753 2-0891 2-0948

CL 10826 1-0759 10740 10795 11124

T&sx. 1. Forces on a circular cylinder at Fr = 1-0.

Havelock's exact solution is actually a truncated series. The terms of the series are;

CR = l.4461448+0.45294412+0-1781419+002721836-000965735. CL = 078984664+ 0-2473S) +0.05410224-0-00040036+002143521. 1-0

05

05 0

Potential flow about two-dimensional ltydrofoils 123

seem to be converging to a value displaced 3 % from that calculated by using

Havelock's formulas. This difference is probably due to inaccuracies in Havelock's values.

A recent attempt to test thin hydrofoil theory was carried out in Norway by

Walderhaug (1964). A series of experimental tests was undertaken to determine

0 Upper surface

Present method

--Thin hydrofoil theory

Experimental data I I 02 0-4 0-6 0-8 10 (a) 1-0 0-5 05 0-5 0 0-5 1.0

00-2

0-4 0-6 0-8 10 xli 1-0 0 0-2 0-4 0-6 08 1-0 xli (b) (c)

Fxava 7. Comparison of the present method with thin hydrofoil theory and experimental

data for various chord-line depths aid Froude numbers. (a) h/f = 0-3, Fr = 0-774, CL = 0-02,

= 0-0072. (b) h/f= 1-0, Fr = 1414, CL = 0-206, CR = 0-0058. (c) h/f= 20, Fr = 2-0,

CL = 0-273, CR = 00041.

Present method

- - Thin hydrofoil heory o Experimental data

WA

-

Present method

- - Thin hydrofoil theory o Experimental data

d

124 Joseph P. (Jiesing and A M. 0. Smith.

pressure distributions on a 3 % thick plate hydrofoil with an elliptic nose and

trailing edge and a NACA 1.75-65 mean-line camber. This model was designed to closely approximate the camber-line hydrofoil model of Walderhaug's thin hydro-foil theory. Figures 7(a), (b) and (c) present Walderhaug's experimental and

theo-retical results and eenespon1ug values calculated by the present method. Even for this case, the thin hydrofoil theory does not display the qualitative nature

of the experimental pressure distribution. The results of the present method show small pressure peaks at the 10% chord and 90% chord locations. These are caused

by a discontinuity in surface curvature at the points where the elliptic leading and trailing edges join the parallel sides of the hydrofoil.

-20 -1-5 -1-0 -05 0 0-5 1-0 - Present method ---Nishiyama 0 0-2 0-4 0-6 0-8 10

PIomE 8. Comparison of the present method with the theory of Nishiyama (1957) for a NACA 4412 hydrofoil operating at Fr = FO, a = 5°, and mid.chord depth h/f = 1.0. Nishiyama (1957) has developed a theory for thick hydrofoils. In this theory

the boundary condition on the body is satisfied by conformal mapping techniques. Figure 8 shows a comparison of calculations by the present method and calcula-tions by Nishiyama for a 4412 hydrofoil. The pressure distribucalcula-tions are qualita-tively different. An experimental investigation of the same airfoil was carried out by Ausman (1954); some of the results are presented in figures 9(a) and (b). The

conditions under which the experimental values were taken for figure 9(a) are iiearly the same s those for figure 8, and therefore the pressure distributions

should be in qualitative agreement. Comparison of these figures verifies this kind

of agreement between the experimental pressure distribution and the pressure distribution calculated by the present method. The reason for the discrepancy between the latter and Nishiyama's results is not known.

Also shown in figures 9(a) and (b) are results calculated by the present method for the exact conditions of the experiment. The experimental values lie below the

potential or calculated values at equal values of angle of attack. This is to be

0 02 4 06 08 10

(a) (b)

FIGURE9. Comparison of calculated and experimental pressure distributions on a NACA 4412

hydrofoil at equal angles of attack and at equal lift coefficients for various mid chord depths

at Fr = 103. (a) h/e= 094 a 5°, Cj = 058. (b) h/f= 060, a 50,CL= O357.

The experiments conducted by Ausman were undertaken to show that the pressure on the upper surface of a hydrofoil is governed by hydrostatic conditions

in addition to hydrodyimmic conditions. Subsequent to 4usman's experiments, Laitone (1954) published a theory consistent with Ausman's experimental data

that suggests that the minimum pressure on the hydrofoil is related to the

maximum depth that can be produced by a hydraulic jump. Specifically, the

theory states that the minimum pressure coefficient can be no less than

- (h/1)/Fr2. Laitone assumes that the height of the free surface, reduced by

fluid flow over the shallow hydrofoil, is restored to its original height only by a

hydraulic jump. Parkin, Perry & Wu (1955) have shown this to be true only at low Froude numbers and shallow depths. The hydrodynamical theory loses accuracy

when a hydraulic jump appears above the hydrofoil, because the free-surface

boundary has taken on a highly non-linear shape in the immediate vicinity of the hydrofoil. Since the hydrofoil is very near the free surface, the errors induced by

the free-surface non-linearity have no chance to decay with depth.

Figures 10(a) and (b) show to what extent the hydrodynamical theory holds

even when the depth and Froude number are small. Figure 10(c) shows two cases

where a hydraulic jump has occurred above the hydrofoil. The hydrodynamic

theory as calculated by the present method is inaccurate for this case, as is to be expected. The limiting negative pressures for the cases presented in figure 10(c), as

-15 -05 05 0 02

(T

04 &6 08 10 xlP

Present method Experimental

a=5° data - CL= 0586' o U

''-H

iii"

Present --CL=O357 method

--I

Experimental data o

Potential flow about two-dimeñ8ional hyolrofoils 125

thus the minimum pressure. Also shown in figures 9(a) and (b) are the more con-ventional comparisons at equal values of lift. As is to be expected, the agreement between the calculated and experimental results is better when the two are

compared at equal values of lift. 20 -F5 -10 -05 0 05 10

126 Joseph P. Gie.sing and A. M. 0. Smith

calculated according to the theory of Laitone, 'are - 6 92 and - 11.8 for Froude

numbers of 0604 and 0462, respectively. These numbers seem to bear little relation to the pressure distribution except that the experimental results do indeed lie below these limiting values. In figure 10, only the upper surface pre are shown, since they are all that were measured. The data were taken

05

(a)

ml,-'

Fr= 0604

- - Present method

-o-o- Experimental data

15

10

05

02 04 06 08 1 0 x/( 0 02 04 06 08_sit10 (c)

Pioui 10. Comparison of calculated and experimental pressure distributions on the upper

surface of a 12% thick symmetric Joukowaki hydrofoil at 50 angle of attack. (a) Fr = 095,

several values of h/f. (b) h/f = 020, two values of Fr. (c) h/f = 025, two values of Fr. from the experimental work of Parkin et at. (1955) for a 12 % thick symmetric Joukowski hydrofoil. Figure 10(a) exhibits one peculiarity. Near the hydrofoil

nose there is a flattening of the experimental pressure peaks, for all cases, that can-not be explained by hydrostatic effects, viscous effects, or cavitation effects. This

flattening persists even to the depth of 18 chords, which, for practical purposes,

is close to infinite depth. The analytic result is shown in figure 10(a) for infinite

£0OPresent methodExperimental data

\

Present method ooExperimental data = &617 Fr= G989 20 - 15 -10

05

-425 0 025 05

-30 -20 Cp 0 -l001..0 xli -30 -20 -10 0

-l.00.0

xli.

First cylinder only-.

Free surface both cylinders-...

U0, -80 -&0 Cp40

(It

irFr2l

FIGtTBE11. The pressure distributions and wave systems for two tandem circular cylinders

operating at Fr = 20, centre depths = 2radii, C; = 033, C!; = -03, = 003, C; = -0677, C; = -059.

-20 ,I Y0

Upper surface

Free surface with channel

Lower surface Without channel

Channel floor

-C 0

-'-

/

1.0: -3-2-10

1 2 3 4 5 6 7 8 9

FIGua 12. The pressure distribution and wave system for a circular cylinder operating at Fr = 080 in a channel. Also shown is the pressure distribution along the channel floor.Circle

centre depth = 2 radii, channel floor depth = 5 radii, CL = 204, CR = ll9.

Figures 11 and 12 present two additional examples,. of multiple bodies. Figure 11 presents the pressure distribution and linear free-surfacedisplacement

for two circular cylinders of unit radius in tandem. The free-surfacedisplacement

for a single circle is also shown.. The drag of the system of two circles is

approxi-mately one-tenth the drag of a single circle. This reduction in drag is caused by

a cancellation of the trailing-wave system of the first circle by that of the second.

Pot entia flow, about two-dimensional hydrofoil.s 1.27

depth. It can be seen that the calculation made by the present method for a

traiHng edge depth of 18 chords is in better agreement with the analytic result than the experimental result is.

Further eXample8

In order to illustrate the capabilities of the present method, severalcalculations involving multiple bodies are presented. One example, the airfoil with slot and flap shown in figure 4, has already been mentioned. In the example the depth is

128 _{Josep7i P. Giesing and A. If. 0. Smit1}

The distance between the two circles was selected as the distance between two

dipoles whose wave trains cancel to zero.

Figure 12 presents the pressure distribution and linear free-surface

displace-nient for a circle in a channel of finite depth. Since the present method was ceveloped for a fluid unbounded in depth, the channel was simulated by simply placing a plane wall, 80 radii in length, as a second body in the infinitely deep

fluid. The figure also shows that the wave effects have been reducedby the depth to the extent that the pressure distribution over the channel floor doesnot show the presence of the waves.

1

2

3

Fianax 13. Streamlines for a circular cylinder operating at Fr= 20, centre depth = 2 radii.

Linearized free surface

-155

T

Linearized free surface

.-i_

Free surface streamline

1' = 0 Free surface streamline

Y= 0

---mjn 14. Streamine for a circular cylinder operating at Fr = 1.0;

centre depth = 2 radii.

As is stated in the theory, the velocity at any point can be determined once the

surface source strength and cyclic constants are known. Figure 13 presents the streamline pattern developed by a circular cylinder under a free surface. The streamlines were obtained by numerical integration of the velocity field. It can

seen that the free-surface streamline does not correspond exactly to the linear

-surface displacement. In extreme cases the streamlinepattern_{may become}

quite unrealistic, as was recently shown by Tuck (1965). Tuck plotted_the

stream-lines for a dipole under a free surface. In this case the dipole represents only an

approximation of a circular cylinder. The same calculation was made for a

JIfiow aloug $wo-dinnsionaö hydrofoils

129 circular cylinder by the present method, and analogous results were obtained

(see figure 14).

Work presented here was conducted by the Douglas Aircraft Company, Inc.,

Aircraft Division, under company-sponsored Research and Development Funds. REFERENCES

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GiEsuco, J. P. 1966 Two-dimensional airfoil methods. Douglas Aircraft Company Rept.

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HARIttS, T. A. & LowRy, é. 0. 1942 Pressure distribution over an NACA 23012 Airfoil with

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WEHAUSEN, 3. V. & Lsrro, E. V. 1960 Handbook of Physics, 9, Surface Waves. Berlin:

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